Three-dimensional Convex Research Articles

Monte Carlo simulation and dosimetric verification of radiotherapy beam modifiers

Monte Carlo simulation of beam modifiers such as physical wedges and compensating filters has been performed with a rectilinear voxel geometry module. A modified version of the EGS4/DOSXYZ code has been developed for this purpose. The new implementations have been validated against the BEAM Monte Carlo code using its standard component modules (CMs) in several geometrical conditions. No significant disagreements were found within the statistical errors of 0.5% for photons and 2% for electrons. The clinical applicability and flexibility of the new version of the code has been assessed through an extensive verification versus dosimetric data. Both Varian multi-leaf collimator (MLC) wedges and standard wedges have been simulated and compared against experiments for 6 MV photon beams and different field sizes. Good agreement was found between calculated and measured depth doses and lateral dose profiles along both wedged and unwedged directions for different depths and focus-to-surface distances. Furthermore, Monte Carlo-generated output factors for both open and wedged fields agreed with linac commissioning beam data within statistical uncertainties of the calculations (<3% at largest depths). Compensating filters of both low-density and high-density materials have also been successfully simulated. As a demonstration, a wax compensating filter with a complex three-dimensional concave and convex geometry has been modelled through a CT scan import. Calculated depth doses and lateral dose profiles for different field sizes agreed well with experiments. The code was used to investigate the performance of a commercial treatment planning system in designing compensators. Dose distributions in a heterogeneous water phantom emulating the head and neck region were calculated with the convolution–superposition method (pencil beam and collapsed cone implementations) and compared against those from the MC code developed herein. The new technique presented in this work is versatile, DICOM-RT compliant and accurate in the simulation of beam modulators. This paper addresses the need to reduce the sources of error in the modelling of beam modifiers since they remain a viable alternative to the MLC technique in the delivery of IMRT beams.

Three-dimensional quantitative structure–activity relationship for several bioactive peptides searched by a convex hull–comparative molecular field analysis approach

Three-dimensional (3D) convex hulls are computed for theoretically generated structures of a group of 18 bioactive tachykinin peptides. The number of peptides treated as a training set is 14, whereas that treated as a test set is four. The frequency of atoms of the same atomic type lying at the vertices of all the hulls computed for all the structures in a structural set is counted. Vertex atoms with non-zero frequency counted are collected together as a set of commonly exposed groups. These commonly exposed atoms are then treated as a set of correspondences for aligning all the other 13 structures in a structural set against a common template, which is the structure of the most potent peptide in the set using the FIT module of the sybyl 6.6 program. Each aligned structural set is then analyzed by the comparative molecular field analysis (CoMFA) module using the C.3 probe having a charge of +1.0. The corresponding cross-validated r 2 values range from −0.99 to 0.57 for a number of 73 structural sets studied. The comparative molecular similarity indices analysis (CoMSIA) module within the sybyl 6.6 package is also used to analyze some of these aligned structural sets. Although the CoMSIA results are in accord with those of CoMFA, it is also found that the CoMFA results of several structural sets can be improved somewhat for conformations of the structures in the sets that are adjusted by constraint energy minimization and then constraint molecular dynamics simulation runs using distance constraints derived from some commonly exposed groups determined for them. This result further implies that the convex hull–CoMFA is a feasible approach to screen the bioactive conformations for molecules of this class.

Relationships between widths of a convex body and of an inscribed parallelotope

Assume that a parallelotope P is inscribed in a three-dimensional convex body C A conjecture says that , where Wi is the ratio of the width of C to the width of P for the direction perpendicular to the i-th pair of parallel facets of P. We prove three weaker inequalities. One of them is , where a3 denotes the related axial diameter of C.

Classification of Some Active Compounds and Their Inactive Analogues Using Two Three-Dimensional Molecular Descriptors Derived from Computation of Three-Dimensional Convex Hulls for Structures Theoretically Generated for Them

Two three-dimensional (3D) molecular descriptors are used to classify 73 protease inhibitors against the human immunodeficiency virus type 1 (HIV-1). X-ray structures of these HIV-1 protease bound inhibitors are used as templates to generate the most probable bioactive conformations of the inhibitors. A convex hull computation algorithm is applied to each structure generated. The frequency of atoms lying on the vertexes of each hull is counted. Vertexes of the same atomic charge state are then gathered together as a set of commonly exposed groups for all the structures generated. The first 3D descriptor is computed as the maximum molecular path length among any three distinct commonly exposed groups, while the second 3D one is computed as the maximum molecular path length among any three atoms of nonconvex hull vertexes. We find that the 73 HIV-1 protease inhibitors can be classified by the first 3D descriptor into two groups, which agrees with the result of visual classification using the activity data as a criterion for these compounds. The classification scheme is then used to classify a database of 427 active trypsin inhibitors and their inactive analogues. The structures of these compounds are generated theoretically from steps of energy minimization and molecular dynamics. Classification for all these compounds is performed using the SYBYL hierarchical clustering method on the first 3D descriptor and then the second 3D one computed. It is found that some inactive analogues are completely separated from the active inhibitors at the first stage of classification using the first 3D descriptor. Most of the highly active inhibitors are classified into a cluster at the second stage of classification using the second 3D descriptor. Finally, most of these highly active inhibitors are separated from all the accompanying inactive analogues in the cluster through a structural alignment process using a set of commonly exposed groups determined for them.

Affine images of the rhombo-dodecahedron that are circumscribed about a three-dimensional convex body

The main result of the paper is dual to the author's earlier theorem on the affine images of the cube-octahedron (the convex hull of the midpoints of edges of a cube) inscribed in a three-dimensional convex body. The rhombododecahedron is the polyhedron dual to the cube-octahedron. Let us call a convex body in Κ⊂ℝ3 exceptional if it contains a parallelogram P and is contained in a cylinder with directrix P. It is proved that any nonexceptional convex body is inscribed in an affine image of the rhombo-dodecahedron. The author does not know if the assertion is true for all three-dimensional convex bodies. Bibliography: 2 titles.

Lattice Points in Three-Dimensional Convex Bodies

Lattice Points in Three-Dimensional Convex Bodies

Three-dimensional convex hull as a fruitful source of diagrams

The relationship between a three-dimensional convex hull and many important two-dimensional diagrams is surveyed, and a unifying approach to robust software for those diagrams is proposed. In this approach, instead of implementing individual algorithms, we first construct a robust algorithm for the three-dimensional convex hull, and next use it as an oracle for computing the other diagrams. The diagrams computable by this approach include the Voronoi diagram and its farthest-point versions both on the plane and on the sphere, the Laguerre Voronoi diagram and its farthest-point version both on the plane and on the sphere, and the intersection of half-spaces.

Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes

This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P .

Computing the width of a three-dimensional point set

We describe a robust, exact, and efficient implementation of an algorithm that computes the width of a three-dimensional point set. The algorithm is based on efficient solutions to problems that are at the heart of computational geometry: three-dimensional convex hulls, point location in planar graphs, and computing intersections between line segments. The latter two problems have to be solved for planar graphs and segments on the unit sphere, rather than in the two-dimensional plane. The implementation is based on LEDA, and the geometric objects are represented using exact rational arithmetic.

Direct method for numerical study of radiation trapping

A direct method for numerical investigation of radiation trapping within the framework of the Holstein-Biberman equation is put forward. The method is based on time propagation of the initial distribution of excited atoms with the split-propagation technique. Some results obtained for circular and elliptic geometries of the gas cells are given in comparison with the data of previous calculations. The present method allows one to investigate various transfer phenomena occurring in an arbitrary (three-dimensional) convex spatial region and driven by the external conditions with arbitrary time dependences.

Comparative noninformativities of quantum priors based on monotone metrics

We consider a number of prior probability distributions of particular interest, all being defined on the three-dimensional convex set of two-level quantum systems. Each distribution is — following recent work of Petz and Sudar — taken to be proportional to the volume element of a monotone metric on that Riemannian manifold. We apply and entropy-based test (a variant of one recently developed by Clarke) to determine which of two priors is more noninformative in nature. This involves converting them to posterior probability distributions based on some set of hypothesized outcomes of measurements of the quantum system in question. It is, then, ascertained whether or not the relative entropy (Kullback-Leibler statistic) between a pair of priors increases or decreases when one of them is exchanged with its corresponding posterior. The findings lead us to assert that the maximal monotone metric yields the most noninformative prior distribution and the minimal monotone (that is, the Bures) metric, the least. Our conclusions both agree and disagree, in certain respects, with ones recently reached by Hall, who relied upon a less specific test criterion than our entropy-based one.

A particle transport problem with non-homogeneous reflection boundary conditions

We consider particle transport in a three-dimensional convex region V, bounded by the regular surface ∂V. We assume that particles are specularly reflected by ∂V and that a source q is assigned on ∂V; more general non-homogeneous boundary conditions are also discussed. The problem is non-linear because the boundary condition is not homogeneous. We prove existence of a unique strict solution and by using the theory of semigroups we derive the explicit expression of such a solution in terms of the boundary source q. In the appendix, we indicate how some properties of affine operators can be used to derive the solution. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

Probing the curvature of a convex object via discrete active touch

In this paper we propose a method of probing the local curvature of a three-dimensional (3D) convex object using the force and torque information which is obtained from the discrete active touch with a robot finger. A scheme for estimating two-dimensional (2D) curvature of a convex object is introduced and the way of extracting 3D curvatures from 2D curvatures is presented. To verify the effectiveness of the proposed method, numerical simulations have been performed and experiments have also been carried out with an experimental system consisting of a finger.

Topology-oriented construction of three-dimensional convex hulls

The divide-and-conquer algorithm for constructing three-dimensional convex hulls is implemented in a numerically robust manner. The approach taken here is a topology-oriented approach, in which the topological consistency is considered more important than the result of numerical computation. This is the first implementation of the divide-and-conquer algorithm for the three-dimensional convex hull using the topology-oriented approach. Implementation details are described together with computational experiments.

Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ d are defined up to a translation, and the relevant statistic is their perimeter.

A Randomized Parallel Three-Dimensional Convex Hull Algorithm for Coarse-Grained Multicomputers

We present a randomized parallel algorithm for constructing the three-dimensional convex hull on a generic p-processor coarse-grained multicomputer with arbitrary interconnection network and n/p local memory per processor, where n/p ? p 2+? (for some arbitrarily small ? > 0). For any given set of n points in 3-space, the algorithm computes the three-dimensional convex hull, with high probability, in $O((n\log{n})/p)$ local computation time and O(1) communication phases with at most O(n/p) data sent/received by each processor. That is, with high probability, the algorithm computes the three-dimensional convex hull of an arbitrary point set in time $O((n\log n)/{p} + \Gamma_{n,p})$ , where Γ n,p denotes the time complexity of one communication phase. The assumption n/p ? p 2+? implies a coarse-grained, limited parallelism, model which is applicable to most commercially available multiprocessors. In the terminology of the BSP model, our algorithm requires, with high probability, O(1) supersteps, synchronization period $L = \Theta((n\log n)/{p})$ , computation cost $O((n\log n)/{p})$ , and communication cost O((n/p) g).

Star Unfolding of a Polytope with Applications

We introduce the notion of a star unfolding of the surface ${\cal P}$ of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on ${\cal P}$. The first algorithm computes the edge sequences traversed by shortest paths on ${\cal P}$ in time $O(n^6 \beta (n) \log n)$, where $\beta (n)$ is an extremely slowly growing function. A much simpler $O(n^6)$ time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an $O(n^{8}\log n)$ time procedure for computing the geodesic diameter of ${\cal P}$: the maximum possible separation of two points on ${\cal P}$ with the distance measured along ${\cal P}$. Finally, we describe an algorithm that preprocesses ${\cal P}$ into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter $1 \le m \le n^2$, it can preprocess ${\cal P}$ in time $O(n^6 m^{1+\delta})$, for any $\delta > 0$, into a data structure of size $O(n^6m^{1+\delta})$, so that a query can be answered in time $O((\sqrt{n}/m^{1/4}) \log n)$. If one query point always lies on an edge of ${\cal P}$, the algorithm can be improved to use $O(n^5 m^{1+\delta})$ preprocessing time and storage and guarantee $O((n/m)^{1/3} \log n)$ query time for any choice of m between 1 and n.

Upper-Bound Elemental Technique (UBET) with convex circular parallelepiped and convex spherical elements for three-dimensional forging analysis

In this study, two kinematically admissible velocity fields are derived for the proposed three-dimensional convex circular parallelepiped and convex spherical UBET elements. Those elements are applied to three-dimensional closed-die forging having convex curve surfaces; the capability of the proposed elements is then demonstrated. From the derived velocity fields, the upper-bound loads on the upper die and the velocity field are determined by minimizing the total energy consumption with respect to some chosen parameters. Also, experiments of two closed-die forgings are performed with commercial pure lead billets at the ambient temperature. The theoretical predictions of the forming load correlated well with the experimental results. The results in this study confirm that the elements proposed in this work can effectively be used to predict the forming load accurately in three-dimensional closed-die forging with convex curve surfaces.

On the girth of convex bodies

For a three-dimensional convex body and a given direction the corresponding girth is defined as the perimeter of the orthogonal projection of the body onto a plane orthogonal to the assigned direction. Obvious examples show that there are pairs of convex bodies that have in every direction equal girths but are not translates of each other. Here a modification of the concept of the girth, called semi-girth, is introduced, and it is shown that convex bodies are uniquely determined (up to translations) by their semi-girths. In addition, corresponding stability results are proved and an inequality concerning the central symmetry of convex domains is established.

Three-dimensional self-avoiding convex polygons

Three-dimensional self-avoiding convex polygons